Calculus Examples

$f (x) = x^{3} + 6 x^{2} + 9 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} + 6 x^{2} + 9 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [9 x]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x)$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 3 x^{2} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 3 x^{2} + 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (9 x)$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 3 x^{2} + 6 (2 x) + \frac{d}{d x} (9 x)$

Step 1.1.2.3

Multiply $2$ by $6$ .

$f' (x) = 3 x^{2} + 12 x + \frac{d}{d x} (9 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 1.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} + 12 x + 9 \frac{d}{d x} (x)$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 3 x^{2} + 12 x + 9 \cdot 1$

Step 1.1.3.3

Multiply $9$ by $1$ .

$f' (x) = 3 x^{2} + 12 x + 9$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 12 x + 9$ .

$3 x^{2} + 12 x + 9$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 12 x + 9 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} + 12 x + 9 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $3$ out of $3 x^{2} + 12 x + 9$ .

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Step 2.2.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 12 x + 9 = 0$

Step 2.2.1.2

Factor $3$ out of $12 x$ .

$3 (x^{2}) + 3 (4 x) + 9 = 0$

Step 2.2.1.3

Factor $3$ out of $9$ .

$3 x^{2} + 3 (4 x) + 3 \cdot 3 = 0$

Step 2.2.1.4

Factor $3$ out of $3 x^{2} + 3 (4 x)$ .

$3 (x^{2} + 4 x) + 3 \cdot 3 = 0$

Step 2.2.1.5

Factor $3$ out of $3 (x^{2} + 4 x) + 3 \cdot 3$ .

$3 (x^{2} + 4 x + 3) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.2.2.1.2

Write the factored form using these integers.

$3 ((x + 1) (x + 3)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$3 (x + 1) (x + 3) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x + 3 = 0$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

The final solution is all the values that make $3 (x + 1) (x + 3) = 0$ true.

$x = - 1, - 3$

Step 3

The values which make the derivative equal to $0$ are $- 1, - 3$ .

$- 1, - 3$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 3) \cup (- 3, - 1) \cup (- 1, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = 3 {(- 4)}^{2} + 12 (- 4) + 9$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 4$ to the power of $2$ .

$f' (- 4) = 3 \cdot 16 + 12 (- 4) + 9$

Step 5.2.1.2

Multiply $3$ by $16$ .

$f' (- 4) = 48 + 12 (- 4) + 9$

Step 5.2.1.3

Multiply $12$ by $- 4$ .

$f' (- 4) = 48 - 48 + 9$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $48$ from $48$ .

$f' (- 4) = 0 + 9$

Step 5.2.2.2

Add $0$ and $9$ .

$f' (- 4) = 9$

Step 5.2.3

The final answer is $9$ .

$9$

Step 5.3

At $x = - 4$ the derivative is $9$ . Since this is positive, the function is increasing on $(- \infty, - 3)$ .

Increasing on $(- \infty, - 3)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 3, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = 3 {(- 2)}^{2} + 12 (- 2) + 9$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 2$ to the power of $2$ .

$f' (- 2) = 3 \cdot 4 + 12 (- 2) + 9$

Step 6.2.1.2

Multiply $3$ by $4$ .

$f' (- 2) = 12 + 12 (- 2) + 9$

Step 6.2.1.3

Multiply $12$ by $- 2$ .

$f' (- 2) = 12 - 24 + 9$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $24$ from $12$ .

$f' (- 2) = - 12 + 9$

Step 6.2.2.2

Add $- 12$ and $9$ .

$f' (- 2) = - 3$

Step 6.2.3

The final answer is $- 3$ .

$- 3$

Step 6.3

At $x = - 2$ the derivative is $- 3$ . Since this is negative, the function is decreasing on $(- 3, - 1)$ .

Decreasing on $(- 3, - 1)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = 3 {(0)}^{2} + 12 (0) + 9$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = 3 \cdot 0 + 12 (0) + 9$

Step 7.2.1.2

Multiply $3$ by $0$ .

$f' (0) = 0 + 12 (0) + 9$

Step 7.2.1.3

Multiply $12$ by $0$ .

$f' (0) = 0 + 0 + 9$

Step 7.2.2

Simplify by adding numbers.

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Step 7.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 9$

Step 7.2.2.2

Add $0$ and $9$ .

$f' (0) = 9$

Step 7.2.3

The final answer is $9$ .

$9$

Step 7.3

At $x = 0$ the derivative is $9$ . Since this is positive, the function is increasing on $(- 1, \infty)$ .

Increasing on $(- 1, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 3), (- 1, \infty)$

Decreasing on: $(- 3, - 1)$

Step 9